In the previous sections, the orders
\(d=(d_1, \dots , d_n)\)  of the species concentrations
are chosen arbitrary. However, theories of singular perturbations
and normally hyperbolic invariant manifolds imply that after a fast transient period,
the dynamics of CRNs is confined
to one low dimensional normally hyperbolic
invariant manifold [1]}, [2]}, where it remains for
a long period, after which it switches to another invariant manifold eventually. Normally hyperbolic invariant manifolds
generalize the notion of hyperbolic fixed points [3]}. Although normally hyperbolic manifolds can have both contracting and expanding directions, here we are only concerned with the fully attractive case. Invariant manifolds with
expanding directions, such as saddle connections, are important for switching between attractive invariant manifolds, a phenomenon that will not be addressed in this paper.
